Lectures on Fractal Geometry and Dynamics

Lectures on fractal geometry and dynamics Michael Hochman∗ March 25, 2012 Contents 1 Introduction 2 Preliminaries 23<ul style="display: flex;"><li style="flex:1">3 Dimension </li><li style="flex:1">3</li></ul>349 3.1 A family of examples: Middle-α Cantor sets . . . . . . . . . . . . . . . . 3.2 Minkowski dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <ul style="display: flex;"><li style="flex:1">4 Using measures to compute dimension </li><li style="flex:1">13 </li></ul>4.1 The mass distribution principle . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Billingsley’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Frostman’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4 Product sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 <ul style="display: flex;"><li style="flex:1">5 Iterated function systems </li><li style="flex:1">24 </li></ul>5.1 The Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.3 Self-similar sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.4 Self-affine sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 <ul style="display: flex;"><li style="flex:1">6 Geometry of measures </li><li style="flex:1">39 </li></ul>6.1 The Besicovitch covering theorem . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Density and differentiation theorems . . . . . . . . . . . . . . . . . . . . 45 6.3 Dimension of a measure at a point . . . . . . . . . . . . . . . . . . . . . 48 6.4 Upper and lower dimension of measures . . . . . . . . . . . . . . . . . . 50 ∗Send comments to [email protected] 1 6.5 Hausdorff measures and their densities . . . . . . . . . . . . . . . . . . . 52 1 Introduction Fractal geometry and its sibling, geometric measure theory, are branches of analysis which study the structure of “irregular” sets and measures in metric spaces, primarily Rd. The distinction between regular and irregular sets is not a precise one but informally, regular sets might be understood as smooth sub-manifolds of Rk, or perhaps Lipschitz graphs, or countable unions of the above; whereas irregular sets include just about everything else, from the middle-13 Cantor set (still highly structured) to arbitrary Cantor sets (irregular, but topologically the same) to truly arbitrary subsets of Rd. For concreteness, let us compare smooth sub-manifolds and Cantor subsets of Rd. These two classes differ in many aspects besides the obvious topological one. Manifolds possess many smooth symmetries; they carry a natural measure (the volume) which has good analytic properties; and in most natural examples, we have a good understanding of their intersections with hyperplanes or with each other, and of their images under linear or smooth maps. On the other hand, Cantor sets typically have few or no smooth symmetries; they may not carry a “natural” measure, and even if they do, its analytical properties are likely to be bad; and even for very simple and concrete examples we do not completely understand their intersections with hyperplanes, or their images under linear maps. The motivation to study the structure of irregular sets, besides the obvious theoretical one, is that many sets arising in analysis, number theory, dynamics and many other mathematical fields are irregular to one degree or another, and the metric and geometric properties of these objects often provides meaningful information about the context in which they arose. At the simplest level, the theories of dimension provide a means to compare the size of sets which coarser notions fail to distinguish. Thus the set of well approximable numbers x ∈ R (those with bounded partial quotients) and the set of Liouvillian numbers both have Lebesgue measure 0, but set of well-approximable numbers has Hausdorff dimension 1, hence it is relatively large, whereas the Liouvillian numbers form a set of Hausdorff dimension 0, and so are “rare”. Going deeper, however, it turns out than many problems in dynamics and number theory can be formulated in terms of bounds on the dimension of the intersection of certain very simple Cantor sets with lines, or linear images of products of Cantor sets. Another connection to dynamics arises from the fact that there is often an intimate relation between the dimension of an invariant set or measure and its entropy (topological or measure-theoretic). Geometric properties may allow us to single out physically significant invariant measures among the many invariant measures of a system. Finer information encoded in an invariant mea- 2sure may actually encode the dynamics which generated it, leading to rigidity results. The list goes on. Our goal in this course is primarily to develop the foundations of geometric measure theory, and we cover in detail a variety of classical subjects. A secondary goal is to demonstrate some applications and interactions with dynamics and metric number theory, and we shall accomplish this mainly by our choices of methods, examples, and open problems which we discuss. We assume familiarity with the basic results on metric spaces, measure theory and Lebesgue integration. 2 Preliminaries N = {1, 2, 3 . . .}. We denote by Br(x) the closed ball of radius r around x: Br(x} = {y : d(x, y) ≤ r} The open ball is denoted Br◦(x); as our considerations are rarely topological is will appear less often. We denote the indicator function of a set A by 1A. We work in Rd or sometimes a complete metric space, and all sets are assumed to be Borel, and all functions are Borel measurable, unless otherwise stated. Also, all measures are Radon unless otherwise stated: recall that µ is Radon if it is a Borel measure taking finite values on compact sets. Such measures are regular, i.e. µ(E) = inf{µ(U) : U is open and E ⊆ U} = sup{µ(K) : K is compact and K ⊆ E} 3 Dimension The most basic quantity of interest in connection to the small scale geometry of a set in a metric space is its dimension. There are many non-equivalent notions with this name. We shall consider the two main ones, Minkowski (box) dimension and Hausdorff dimension. We give the definitions in general for metric spaces, but most of our applications and some of the results in these sections will already be special to Rd. 3.1 A family of examples: Middle-α Cantor sets Before discussing dimension, we introduce one of the simplest families of “fractal” sets, which we will serve to demonstrate the definitions that follow. 3 Let 0 < α < 1. The middle-α Cantor set Cα ⊆ [0, 1] is defined by a recursive procedure. For n = 0, 1, 2, . . . we construct a set Cα,0 which is a union of 2n closed nintervals, indexed by sequences i = i1 . . . in ∈ {0, 1} and each of length ((1−α)/2)n. To begin let Cα,0 = [0, 1] and I = [0, 1] (indexed by the unique empty sequence). Assuming <ul style="display: flex;"><li style="flex:1">that Cα,n has been defined and is the disjoint union of the 2n closed intervals Ii ...i </li><li style="flex:1">,</li></ul>1nni1 . . . in ∈ {0, 1} , divide each of the intervals into the two subintervals, Ii ...i 0, Ii ...i 1 ⊆Ii ...i which remain after removing from Ii the open subinterval with the same center 1n1n1nas Ii ...i and α times shorter. Finally let 1n[Cα,n+1 =Ii n+1 i∈{0,1} Clearly Cα,0 ⊇ Cα,1 ⊇ . . ., and since the sets are compact, ∞\<ul style="display: flex;"><li style="flex:1">Cα = </li><li style="flex:1">Cα,n </li></ul>n=0 is compact and nonempty. All of the sets Cα, 0 < α < 1 are mutually homeomorphic, since all are topologically Cantor sets (i.e. compact and totally disconnected without isolated points). They all are of first Baire category. And they all have Lebesgue measure 0, since one may verify that Leb(Cαn) = (1 − α)n → 0. Hence none of these theories can distinguish between them. Nevertheless qualitatively it is clear that Cα becomes “larger” as α → 0, since decreasing α results in removing shorter intervals at each step. In order to quantify this one uses dimension. 3.2 Minkowski dimension Let (X, d) be a metric space, for A ⊆ X let |A| = diam A = sup d(x, y) x,y∈A SA cover of A is a collection of sets E such that A ⊆ E∈E E. A δ-cover is a cover such that |E| ≤ δ for all E ∈ E. The simplest notion of dimension measures how many sets of small diameter are needed to cover a set. Definition 3.1. Let (X, d) be a metric space. For a bounded set A and δ > 0 let 4 N(A, δ) denote the minimal size of a δ-cover of A, i.e. k[N(A, δ) = min{k : A ⊆ Ai and |Ai| ≤ δ} i=1 The Minkowski dimension of A is log N(A, δ) Mdim(A) = lim δ→∞ log(1/δ) assuming the limit exists. If not we define the upper and lower dimensions log N(A, δ) Mdim(A) = lim sup δ→∞ log(1/δ) log N(A, δ) Mdim(A) = lim inf δ→∞ log(1/δ) Remark 3.2. . 1. Mdim A = α means that N(A, δ) grows approximately as δ−α as δ → 0; more precisely, Mdim A = α if and only if for every ε > 0, δ−(α−ε) ≤ N(A, δ) ≤ δ−(α+ε) for sufficiently small δ > 0 2. Clearly Mdim ≤ Mdim and Mdim exists if and only if the two are equal. 3. Minkowski dimension is not defined for unbounded sets and may be infinite for bounded sets as well, though we will se that it is finite for bounded sets in Rd. 4. From the definitions it is immediate that N(A, δ) ≤ N(B, δ) when A ⊆ B, consequently, Mdim A ≤ Mdim B and similarly for the upper and lower versions. 5. Fromt he definition it is also clear that if δ < δ0 then N(A, δ) ≥ N(A, δ0). In particular if εk & 0 and εk/εk+1 ≤ C < ∞, then we can compute the limits int he definition of Mdim and its variants along δk. Indeed, for every δ > 0 there is a k = k(δ) such that εk+1 < δ ≤ εk. This implies N(A, εk+1) ≤ N(A, δ) ≤ N(A, εk) 5 The assumpetion implies that log(1/δ)/ log(1/εk(δ)) → 1 as δ → 0, so the inequality above implies the claim after taking logarithms and dividing by log(1/δ), log(1/εk), log(1/εk+1). Example 3.3. . 1. A point has Minkowski dimension 0, since N({x0}, δ) = 1 for all δ. More generally N({x1, . . . , xn}, δ) ≤ n, so finite sets have Minkowski dimension 0. 2. A box B in Rd can be covered by c · δ−d boxes of side δ, i.e. N(B, δ) ≤ cδ−d .Hence dim B ≤ d. 13. If A ⊆ Rd has Mdim A < d then Leb(A) = 0. Indeed, choose ε = (d − Mdim A). 2For all small enough δ, there is a cover of A by δ−(Mdim A+ε) sets of diameter ≤ δ. Since a set of diameter ≤ δ can itself be covered by a set of volume < cδd, we find that there is a cover of A of total volume ≤ cδd · δ−(Mdim A+ε) = cδε. Since this holds for arbitrarily small δ, we conclude that Leb(A) = 0. Equivalently, if A ⊆ Rd and Leb(A) > 0 then Mdim A ≥ d. In particular for a box B we have, using (2), that Mdim B = d. 4. A line segment in Rd has Minkowski dimension 1. A relatively open bounded subset of a plane in R3 has Minkowski dimension 2. More generally any compact k-dimensional C1-sub-manifold of Rd has box dimension k. 5. For Cα as before, Mdim Cα = log 2/ log(2/(1 − α)). Let us demonstrate this. To get an upper bound, notice that for δn = ((1 − α)/2)n the sets Cn are covers of Cα by 2n intervals of length δn, hence N(Cα, δn) ≤ 2n. If δn+1 ≤ αδ < δn then clearly N(Cα, δ) ≤ N(Cα, δn+1) ≤ 2n+1 On the other hand every set of diameter ≤ δ can intersect at most two maximal intervals in Cαn+1, hence 1N(Cα, δ) ≥ · 2n 2so for δn+1 ≤ δ < δn <ul style="display: flex;"><li style="flex:1">(n − 1) log 2 </li><li style="flex:1">log N(Cα, δ) </li><li style="flex:1">(n + 1) log 2 </li></ul><ul style="display: flex;"><li style="flex:1">≤</li><li style="flex:1">≤</li></ul><ul style="display: flex;"><li style="flex:1">(n + 1) log(2/(1 − α)) </li><li style="flex:1">log 1/δ </li><li style="flex:1">n log(2/(1 − α)) </li></ul>and so, taking δ → 0, Mdim Cα = log 2/ log(2/(1 − α)) Proposition 3.4. Properties of Minkowski dimension: 1. Mdim A = Mdim A 62. Mdim A depends only on the induced metric on A. 3. If f : X → Y is Lipschitz then Mdim fA ≤ Mdim A, and if f is bi-Lipschitz then Mdim fA = Mdim A. Proof. By inclusion Mdim A ≤ Mdim A, so for the first claim we can assume that Mdim A < ∞. Then N(A, ε) = N(A, ε) for every ε > 0, because in general if A ⊆ <ul style="display: flex;"><li style="flex:1">S</li><li style="flex:1">S</li></ul>For the second claim, note that the diameter of a set depends only ont he induced <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>i=1 Ai then A ⊆ i=1 Ai, and if {Ai} is a δ-cover then so is {Ai}. This implies the claim. <ul style="display: flex;"><li style="flex:1">S</li><li style="flex:1">S</li></ul>metric, and if A ⊆ Ai then A ⊆ (Ai∩A) and |Ai∩A| ≤ |Ai|, so N(A, ε) is unchanged if we consider only covers by subsets of A. <ul style="display: flex;"><li style="flex:1">S</li><li style="flex:1">S</li></ul>Finally if A ⊆ Ai then f(A) ⊆ f(Ai), and if c is the Lipschitz constant of f then |f(E)| ≤ c|E|. Thus N(fA, cε) ≤ N(A, ε) and the claim follows. The example of the middle-α Cantor sets demonstrates that Mankowski dimension is not a topological notion, since the sets Cα all have different dimensions, but for 0 < α < 1 they are all topologically a Cantor set and therefore homeomorphic. On the other hand the last part of the proposition shows that dimension is an invariant in the bi-Lipschitz category. Thus, Corollary 3.5. For 1 < α < β < 1, the sets Cα, Cβ, are not bi-Lipschitz equivalent, and in particular are not C1-diffeomorphic, i.e. there is no bi-Lipschitz map f : Cα → Cβ. Next, we specialize to Euclidean space. First we note that, although the same topological space can have different dimensions depending on the metric, changing the norm on Rd does not have any effect, since the identity map is bi-Lipschitz, all norms on Rd being equivalent. Second, as we shall see next, in Rd one can compute the Minkowski dimension using covers by convenient families of cubes, rather than arbitrary sets. This is why Minkowski dimension is often called box dimension. Definition 3.6. Let b ≥ 2 be an integer. The partition of R into b-adic intervals is k k + 1 Db = {[ , ) : k ∈ Z} <ul style="display: flex;"><li style="flex:1">b</li><li style="flex:1">b</li></ul>The corresponding partition of Rd into b-adic cubes is dDb = {I1 × . . . × Id : Ii ∈ Db} (We suppress the superscript d when it is clear from the context). The covering number of A ⊆ Rd by b-adic cubes is N(X, Db) = #{D ∈ Db : D ∩ X = ∅} 7Lemma 3.7. For any integer b ≥ 2, 1n<ul style="display: flex;"><li style="flex:1">Mdim X = lim </li><li style="flex:1">log N(X, Db ) </li></ul>n→∞ n log b and similarly for Mdim and Mdim. Proof. Since D ∈ Db has |D| = c · b−n (in fact for the norm k·k∞ the constant is c = 1, nfor other norms it depends on d), we find that N(A, c · b−n) ≤ N(A, Db ) nOn the other hand every set B with |B| ≤ b−n can be covered by at most 2d cubes nD ∈ Db . Hence d −n )nN(A, Db ) ≤ 2 N(A, b Substituting this into the limit defining Mdim, and interpolating for b−n−1 ≤ δ < b−n as in Example 3.3 (5), the lemma follows. Example 3.8. Let E ⊆ N. The upper and lower densities of E are 1d(E) = lim sup |E ∩ {1, . . . , n}| n1n→∞ d(E) = lim inf |E ∩ {1, . . . , n}| n→∞ nLet ∞XXE = { <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">−nxn : xn = 0 if n ∈/ E and xn ∈ {0, 1} otherwise} </li></ul>n=1 We claim that Mdim XE = d(E) and Mdim XE = d(E). Indeed, for each initial P∞ sequence x1 . . . xk, the set of numbers of the form n=1 2−nxn consist of a single level-k dyadic interval plus one point. Thus the number of level-k dyadic intervals needed to cover XE is, to within a factor of 2, equal to the number of sequences x1 . . . xk whose digits satisfy the condition in the definition of XE. The number of such sequences is precisely 2|E∩{1,...,k}|. In summary, we have found that 2|E∩{1,...,n}| ≤ N(XE, Dk) ≤ 2 · 2|E∩{1,...,n}| 1nTaking logarithms and dividing by n, we see that the asymptotics of log N(XE, Dk) 1are the same as of n |E ∩ {1, . . . , n}|, as claimed. In particular, since one easily has sets E⊆ N with d(E) < d(E) we see that the lower

Lectures on Fractal Geometry and Dynamics

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support